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Properties

Label 2-765-5.4-c1-0-29
Degree $2$
Conductor $765$
Sign $0.816 + 0.576i$
Analytic cond. $6.10855$
Root an. cond. $2.47154$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.134i·2-s + 1.98·4-s + (1.29 − 1.82i)5-s − 2.86i·7-s + 0.536i·8-s + (0.245 + 0.173i)10-s + 3.84·11-s + 6.22i·13-s + 0.385·14-s + 3.89·16-s − i·17-s − 6.62·19-s + (2.55 − 3.61i)20-s + 0.517i·22-s − 4.51i·23-s + ⋯
L(s)  = 1  + 0.0951i·2-s + 0.990·4-s + (0.576 − 0.816i)5-s − 1.08i·7-s + 0.189i·8-s + (0.0777 + 0.0549i)10-s + 1.15·11-s + 1.72i·13-s + 0.103·14-s + 0.972·16-s − 0.242i·17-s − 1.51·19-s + (0.571 − 0.809i)20-s + 0.110i·22-s − 0.942i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.816 + 0.576i$
Analytic conductor: \(6.10855\)
Root analytic conductor: \(2.47154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{765} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 765,\ (\ :1/2),\ 0.816 + 0.576i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09736 - 0.666122i\)
\(L(\frac12)\) \(\approx\) \(2.09736 - 0.666122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.29 + 1.82i)T \)
17 \( 1 + iT \)
good2 \( 1 - 0.134iT - 2T^{2} \)
7 \( 1 + 2.86iT - 7T^{2} \)
11 \( 1 - 3.84T + 11T^{2} \)
13 \( 1 - 6.22iT - 13T^{2} \)
19 \( 1 + 6.62T + 19T^{2} \)
23 \( 1 + 4.51iT - 23T^{2} \)
29 \( 1 + 0.658T + 29T^{2} \)
31 \( 1 - 3.49T + 31T^{2} \)
37 \( 1 + 3.34iT - 37T^{2} \)
41 \( 1 + 2.04T + 41T^{2} \)
43 \( 1 + 1.29iT - 43T^{2} \)
47 \( 1 - 6.22iT - 47T^{2} \)
53 \( 1 - 9.92iT - 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 + 7.61T + 61T^{2} \)
67 \( 1 + 0.257iT - 67T^{2} \)
71 \( 1 + 1.18T + 71T^{2} \)
73 \( 1 - 3.26iT - 73T^{2} \)
79 \( 1 - 4.99T + 79T^{2} \)
83 \( 1 + 7.91iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 4.08iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.33855322283161111029222630801, −9.317878060558961499575708037195, −8.667073361557845471635792401869, −7.49633888442081760630383927092, −6.53702311688600673955239364443, −6.25235606743853251275429731171, −4.61126482439666627031433845377, −4.00115573920311044021990030987, −2.24493872714039761691751471878, −1.27056794307376461034071194360, 1.73276468480461995781299047156, 2.70774650939252506243776706012, 3.57936717098165328086042245843, 5.37998921030838786217479970310, 6.14354007244576628841618845771, 6.67563231815888736036945884215, 7.80208468783366213576616201742, 8.686678639561360744773331600356, 9.772966782973070512234452924276, 10.45360174132583951588833411568

Graph of the $Z$-function along the critical line